The nonvanishing trace conjecture for permutation RSK operators

For each positive integer $d$, let ${\sf RSK}_{1^d,1^d}$ be the RSK operator indexed by the compositions $1^d$, and let ${\sf RSK}_{m,n,d}$ be the corresponding operator in dimensions $m,n$. Nonvanishing trace conjecture.

$$\operatorname{Tr}\\,{f RSK}_{1^d,1^d}\neq 0$$

for $d\neq 2$; equivalently, ${\operatorname{Tr}}\\,{f RSK}_{m,n,d}$ has total degree $2d$ for $d\neq 2$. The computed values through $d=11$ support the claim, but no general proof is supplied.