Conjectural tensor-product formulae relative to M

Let $p>0$ be a prime, let $\Bbbk$ be a field of characteristic $p$, and let $G$ be an elementary abelian $p$-group of order $q=p^n$. Let $W$ be an indecomposable $\Bbbk G$-module of dimension $2$, define $V_i=S^{i-1}(W)^*$ for $i=1,\ldots,q$, and set

$$M:=\bigoplus_{r=0}^{p^{n-1}}V_{rp}.$$

For $i,j<q$ with $i+j\leq q$, write $i=rp+i'$ and $j=sp+j'$ with $i',j'<p$, and assume $i'+j'\leq p$. Then:

Conjectural tensor-product formulae.

$$V_i\otimes V_j\cong_M\bigoplus_{l=1}^{\min(i',j')}V_{i+j-(2l-1)}.$$

Moreover,

$$V_{pr+p-i'}\otimes V_{ps+p-j'}\cong_M V_i\otimes V_j.$$

These are proposed generalisations of the preceding corollaries for tensor products modulo summands projective relative to $M$. The authors state that they were unable to prove these formulae; their status is therefore open.