Finite-exception conjecture for the residual solutions of $s(n^2)=5$

Let $E_5$ be the set of odd positive integers $n$ satisfying $s(n^2)=5$. The paper decomposes this set as

$$E_5=\{1+2+2^\ell:\ell\geq3\}\cup\{1+2^\ell+2^{\ell+1}:\ell\geq3\}\cup\{1+2^\ell+2^{2\ell-1}:\ell\geq3\}\cup E',$$

where $E'$ is finite.

Finite-exception conjecture. The residual finite set is

$$E'=\{29,31,51,79,91,95,157,223,279,479,727,1471,5793\}.$$

The displayed decomposition is a theorem in the paper, but the exact description of the residual set is conjectural and remains open.