Conjecture based on an integer sequence from R.P. Stanley

Question

OEIS sequence A067855 gives the sum of squares of a vector of coefficients of $s_\mu$, $\mu\vdash 2n$, produced by summing the expansion of $s_\lambda^2$ over all partitions $\lambda\vdash n$. It appears to have a simple generating function.
Conjecture:
replacing $s_\lambda^2$ by $s_\lambda s_{\lambda´}$ results in the same sequence. This is not trivial since the vector of coefficients is different. It just 'magically' gives the same length squared.
Is it trivial to prove this?