Positivity conjecture for Sylow branching coefficients of powers of two

Let $k\in\mathbb{N}$ and let $\lambda\vdash 2^k$. Write $l(\lambda)$ for the number of parts of $\lambda$, let $\lambda_{l(\lambda)}$ be its smallest part, and let $Z^\lambda$ be the Sylow branching coefficient defined in the source. Sylow branching positivity conjecture. If $\lambda_{l(\lambda)}\ge 2$, then

$$Z^\lambda>0$$

unless $\lambda=(5,3)$, or $k\ge 3$ and $\lambda=(3,3,2^{2^{k-1}-3})$. This conjecture gives a proposed classification of the exceptional zero Sylow branching coefficients in the stated family; the source does not report a resolution.