Does specific gravity keep increasing after conversion is complete?

Yes it's normal for the SG to rise during a step mash. But will no longer rise once conversion is complete (check with iodine test)

While you can monitor the SG during the mash and it is a useful measurement, it's difficult to be accurate. It really needs to be a recirculating mash (RIS) so it's evenly mixed. Also sparge volumes need to be added in as dilution. Full volume BIAB won't have this issue, but usually you won't have RIS in a simple system. Many will say the uncovered starches skew the readings in refracto meters and even buoyancy tests. While technically a sound caution, it's not what I've experienced and my readings via refractometer are pretty close if you can capture clean wort. Mainly because the starches are not disolved and mostly hang out in the endosperm until converters to sugar and dissolved.

Conversion of starches to sugar is only part of the issue. The iodine test will tell you when the starches are gone, but it won't tell you which sugars are in the solution. A longer mash will allow the longer sugars to be broken down into shorter ones. That may not affect SG but it will affect fermentability. Since a refractometer is adjusted to read simple sugars, it may respond to that change, but a hydrometer would not.

I have found that patience is rewarded when making beer, and I am trying to simplify my process. I no longer use an iodine test and just let it sit for an hour at my desired temperature.

Short answer for your question is no. But I don't really understand what you were measuring during the mash steps. Mash is usually thick until you do a sparge to get the most solids out of your grain. You usually measure your SG after the sparge (pre-boil SG).

Until you know your equipment effectiveness more accurately, this is just about hitting approximate values. But you can easily correct your SG before boiling by just adding water (to lower SG) or boiling out more water (with longer boil time or adding heat power to raise SG). There are some calculators to measure the amount of water needed for this, just search for 'Dilution And Boiloff Gravity Calculator'.

As for the refractometer measurements during the mash steps - don't do that. They are much useless during that period. Just take them after sparge and correct SG if required as noted above.

If you need to know if your conversion is actually complete there is a simple solution for this - iodine test. Just get a iodine tincture (alcohol iodine solution), take a liquid part (without solids of grain) of your wort from the mash and add a tiny drop of iodine. If conversion are done it shouldn't change it's color. If it turns blue of violet this means starch is still present and you must continue with conversion. If it doesn't changes it's color that means the conversion are complete (though some processes still goes on affection your FG/body, that is why mashout is recommended, to stop those). Also if it's red that means the conversion is nearly complete, starch is absent but some of the long dextrins are still in place which is might be actually good for a very full bodied beers. Iodine test is much more accurate if you really want to know if your starch conversion are done.

Tbh I'm not sure what the question is.

Palmer in one of Brew Strong podcasts says that approx 80% of all conversion occurs within the first 20 minutes or so. Longer mash time is required to finish conversion of remaining "long tail". Brulosophy's Marshall Schott also says you can cut total mash time from 60 to 45 minutes without much impact on the taste and gravity.

The modern malts most likely don't require all that step mash black magic. I used to do step mashes, overnight mashes, even a decoction once or twice, but when I switched to single infusions, I didn't see any difference in efficiency, fermentability or taste.

I guess the point of this ramble is.. What I'd do (and, in fact, did). Do 1 hour single infusion mash at 148, then short mashout at 175, measure your system's efficiency, record that parameter in Beersmith (or whatever brewing software you use) and stop worrying about it. That will free up thinking resources to worry about really important stuff like yeast health and water chemistry.