Ideas Behind The Jobava-Prié (part three)

AFTER 1.d4 d5 2.Nc3 Nf6 3.Bf4 a6!? 4.e3 Bf5, White's main alternative to 5.f3 is 5.Bd3, which has been played by Magnus Carlsen and Baadur Jobava, the Georgian grandmaster after whom the opening is partly named.
The idea is firstly to swop off Black's well-placed light-square bishop, but at the same time, at least as played by Carlsen and Jobava, to strengthen White's centre.
Magnus Carlsen (GM 2832) - Hikaru Nakamura (GM 2785)
Norway Masters Blitz (Stavanger) 2017
5...Bxd3
5...Bg6 and 5...e6!? are playable alternatives.
6.cxd3
Quick development with 6.Qxd3 has been played by a 2557, but presumably Carlsen and Jobava feel it is a position in which strengthening the centre is more important.
6...e6 7.Nf3 Bd6 8.Bg5!?
Stockfish10's choice.
Jobava last year chose 8.0-0 against a player rated more than 400 points below him, but could only draw.
8...Nbd7 9.0-0
Stockfish10 much prefers 9.e4!?, but Komodo10 is unsure.
9...h6 10.Bh4 0-0 11.e4
White starts to mobilise his central pawn-majority, which seems his main trump in the position.
11.Be7 12.Qb3 Nb6 13.Rac1 c6 14.Rfe1 Rc8 15.h3 Nfd7
Seeking an exchange of bishops to relieve the cramp in Black's position.
16.Bg3 Re8 17.a3 Bf8 18.Qa2 Nf6 19.b4
Starting a Minority Attack with the aim of giving Black a queenside pawn weakness.
19...Ra8
Black cannot really prevent a4 followed by b5, but if they come he will meet the latter with ...axb5, opening the a file for his repositioned queen's rook.
20.Qb2 Nh5 21.Bh2 Nf6 22.Red1 Qe7
The engines reckon 22...a5 gives Black a good game.
23.Nd2 Rec8 24.Nb3 Nfd7 25.Re1 Qd8 26.Nc5 Ra7 27.a4
After some cat-and-mouse manoeuvring, Carlsen proceeds with his Minority Attack.
27...Nxc5 28.dxc5!?
The engines prefer leaving Black with a backward b pawn on a half-open file after 28.bxc5.
28...Nd7 29.exd5 exd5 30.d4 b6!?
Nakamura immediately seeks to undermine White's queenside space advantage.
31.b5!?
The engines prefer a less-sharp move, such as 31.a5.

How should Black proceed?

*****
*****
*****
*****
*****
31...axb5
Black is slightly better after 31...cxb5!?, according to the engines, eg 32.axb5 (32.c6!?) bxc5 33.bxa6 Nf6 34.Qb5 cxd4 35.Ne2 Qe8 36.Nxd4 Qxb5 37.Nxb5 Rxc1 38.Rxc1 Rxa6, when Black is a pawn up but his pieces are a little uncoordinated.
32.axb5 bxc5 33.bxc6 Rxc6 34.Nxd5 (½–½, 67 moves)