I am guessing it will be closer to the $15.5M APY Nate Solder got with the Giants.
It's possible that the Pats would go a bit higher for an LT who turned 25 (4/13) the day after Solder turned 30 (4/12).
Solder's contract is for four years, $35M guaranteed at signing - which includes a $16M signing bonus and his base salaries for 2018 & 2019. He has "first day" roster bonuses for 2020 ($3M) & 2021 ($4M) on top of $9.9M salaries that aren't guaranteed until week-1 each year.
Jake Matthews (age 26) signed a 5-year, $14.5M APY extension with the Falcons a couple months
after Solder's deal was done. He got $26M guaranteed at signing, including a $14M signing bonus and his base salaries for 2018 & 2019. That made Matthews the 3rd highest-paid LT (by APY) behind Solder and Taylor Lewan ($16M APY), who also sign his deal after Solder's.
After them, Okung, Armstead and Trent Williams have 4-year or 5-year deals with APYs around $13M range. Okung signed in 2017. The Williams and Armstead deals were done in 2015 & 2017, respectively.
The next tier down (~$12M APY) includes Tyron Smith, Bakhtiari, Fisher and Glenn. All of those deals were for 5-years and sign in 2016, except for Smith's. The 'Boys signed Smith to an 8-year, $97.6M extension way back in 2014.
A typical Pats extension offer would land somewhere in the
current 2nd-tier to 3rd-tier range. That's where Mason's extension is (relative to other starting RG APYs), and it's about where Cannon's extension landed (relative to other starting RT APYs) back when it was signed (and when he'd only been full-time starter for half a season). My guess is that the Pats would offer Brown a commensurate extension (2nd-tier for LT) as soon as Scar is sure that he's a keeper (which may have already happened). Their initial offer might go as high as $14M, just below the current 1st-tier, on a 5-year or 6-year extension.
Anyway, it's not clear how generous a rational market would be for a player with really only the one year as a proven starter at LT (assuming that he achieves that). Of course, the market isn't really rational with the Browns having a relative crap-tonne of cap space and the motivation to go for broke while Mayfield is cheap. The Jets (with Darnold) and the Colts also have the 2019 cap space to offer a ludicrous contract to Brown next spring.
If those offers are in the $15.5M+ range (and they might well be), I don't think the Pats will go there.