Steinberg tensor product conjecture for general linear supergroups in characteristic 2

Let $P$ denote the object used to form $\operatorname{GL}(nP)$, and let $\lambda$ be a highest weight of $\operatorname{GL}(nP)$. Call $\lambda$ 4-restricted if

$$\deg\lambda_i-\deg\lambda_{i+1}<4.$$

For a highest $\operatorname{GL}(n)$-weight $\mu=(\mu_1,\dots,\mu_n)$, write $\lambda\cdot\chi^\mu=(\chi^{\mu_1}\otimes\lambda_1,\dots,\chi^{\mu_n}\otimes\lambda_n)$, and let $\operatorname{Fr}^2:\operatorname{GL}(nP)\to\operatorname{GL}(n)$ be the square of the Frobenius homomorphism.

Steinberg tensor product conjecture. For $\lambda$ 4-restricted and $\mu=(\mu_1,\dots,\mu_n)$ a highest $\operatorname{GL}(n)$-weight,

$$L_{\operatorname{GL}(nP)}(\lambda\cdot\chi^\mu)\cong L_{\operatorname{GL}(nP)}(\lambda)\otimes(\operatorname{Fr}^2)^*L_{\operatorname{GL}(n)}(\mu).$$

This proposes the characteristic-2 analogue of the Steinberg tensor product theorem for $\operatorname{GL}(nP)$, using the square of Frobenius because the ordinary Frobenius map does not have the required homomorphism properties in $\operatorname{Ver}_4^+$.