  Dimension conservation for almost every projection has been well-established
by the work of Marstrand, Mattila and Hunt and Kaloshin. More recently, Hochman
and Shmerkin used CP-chains, a tool first introduced by Furstenberg, to prove
all projections preserve dimension of measures on $[0,1]^2$ that are the
product of a $\times m$-invariant and a $\times n$-invariant measure (for $m$,
$n$ multiplicatively independent). Using these tools, Ferguson, Fraser and
Sahlsten extended that conservation result to $(\times m,\times n)$-invariant
measures that are the pushforward of a Bernoulli scheme under the $(m,n)$-adic
symbolic encoding. Their proof relied on a parametrization of conditional
measures which could not be extended beyond the Bernoulli case. In this work,
we extend their result from Bernoulli measures to Gibbs measures on any
transitive SFT. Rather than attempt a similar parametrization, the proof is
achieved by reducing the problem to that of the pointwise convergence of a
double ergodic average which is known to hold when the system is exact.
