Two-step Newell-type plethysm equality

Let $m,n$ be positive integers and let $\nu$ be a partition of $n$. For a partition $\tau$ of $mn$, write $p^{\tau}_{\nu,(m)}$ for the corresponding plethysm coefficient. Two-step Newell-type conjecture. If $\lambda\vdash mn$, then

$$p^\lambda_{\nu,(m)}=p^{\lambda+(2^n)}_{\nu,(m+2)}$$

for any $\nu\vdash n$. The source states that this conjecture is a consequence of the preceding Newell-type conjecture. No resolution is supplied, so it remains open as a separately stated claim.