Warning, really long post.
Let me lay some math on you:
Scenario A: Training naturally, meaning HP is where it should be.
(all players 40 attack, 1 defense, 66.9ish combat)
Player A:
98 str
86 hp
1 prayer
Player B:
96 str
84 hp
13 prayer
Player C:
92 str
80 hp
31 prayer
Strength level when potted + prayer: (note: Runescape always rounds down)
Player A: 110 (98+12)
Player B: 117 (96+12+9)
Player C: 117 (92+12+13)
Conclusion: Doesn't matter if you are 13 or 31 prayer, they are the same.
Scenario B: Soul Wars Prod, you control your HP level
(all players 40 attack, 1 defense, 78 hp, 64.9ish combat)
Player A:
99 Strength
1 Prayer
Player B:
95 Strength
13 Prayer
Player C:
87 Strength
31 Prayer
Strength level when potted + prayer: (note: Runescape always rounds down)
Player A: 111 (99+12)
Player B: 116 (95+12+9)
Player C: 111 (87+11+13)
Conclusion: 13 Prayer is the best for strength levels/combat level.
Final thoughts: Choosing your prayer level depends on where your goals are. If you are only melee, and you are training f2p, then you can pick between 13 or 31, doesn't make a difference. If you train p2p though and keep low hp, 13 prayer is the best. The only drawback of 13 prayer is it's a nuisance to recharge because you can't use the monastery. Also, below ~80 strength, it really doesn't make a difference at all. If you are ranging at all, 44 range prayer is the best period past level ~70 ranged.