Let
p
p
be an odd prime, and
A
n
\mathfrak {A}_n
the alternating group of degree
n
n
. We determine which ordinary irreducible representations of
A
n
\mathfrak {A}_n
remain irreducible in characteristic
p
p
, verifying the author’s conjecture from 2010. Given the preparatory work done in 2010, our task is to determine which self-conjugate partitions label Specht modules for the symmetric group in characteristic
p
p
having exactly two composition factors. This is accomplished through the use of the Robinson–Brundan–Kleshchev ‘
i
i
-restriction’ functors, together with known results on decomposition numbers for the symmetric group and additional results on the Mullineux map and homomorphisms between Specht modules.