The conjecture on Frobenius-twisted unstable-module extensions

Let $\mathcal U$ be the category of unstable modules, let $F(1)$ be the free unstable module generated in degree $1$, and let $\Phi$ denote the Frobenius twist functor. For an even integer $d=2^{n_1}+\cdots+2^{n_k}$ with $n_1<\cdots<n_k$, set $\upsilon(d)=1+n_1-k$. Then, for every integer $d$,

The Frobenius-extension conjecture.

$$\operatorname{Ext}_{\mathcal U}^{d}\left(\Phi^{r}F(1),\Phi^{r}F(1)\right)=\begin{cases}0&\text{if }2\nmid d,\\0&\text{if }2\mid d\text{ and }r<\upsilon(d),\mathbb F_2&\text{otherwise},\end{cases}$$

Moreover, for every $r$ there are monomorphisms

$$\operatorname{Ext}_{\mathcal U}^{d}\left(\Phi^{r}F(1),\Phi^{r}F(1)\right)\hookrightarrow\operatorname{Ext}_{\mathcal U}^{d}\left(\Phi^{r+1}F(1),\Phi^{r+1}F(1)\right).$$

The preceding theorem establishes this behavior in several ranges of $d$ and proves the nontriviality assertion for even $d$ under the stated bound on $r$. The conjecture asks for the formula and injectivity in all degrees.