The F4-type Galois representation counting conjecture

Let $\rho$ be a $26$-dimensional conductor-one geometric $\ell$-adic Galois representation whose image has Zariski closure a connected reductive group of type $\mathrm{F}_{4}$. For $a,b,c,d\in\mathbb{N}$, define its Hodge–Tate multiset by

$$\mathrm{HT}(a,b,c,d)=\left\{0,0,\pm a,\pm b,\pm(a+b),\pm(b+c),\pm(a+b+c),\pm(b+c+d),\pm(a+b+c+d),\pm(a+2b+c),\pm(a+2b+c+d),\pm(a+2b+2c+d),\pm(a+3b+2c+d),\pm(2a+3b+2c+d)\right\}.$$

The F4-type Galois representation counting conjecture. The number of equivalence classes of such representations with $\mathrm{HT}(\rho)=\mathrm{HT}(a,b,c,d)$ and $a,b,c,d\geq 1$ is

$$\mathrm{F}_{4}(a-1,b-1,c-1,d-1),$$

where $\mathrm{F}_{4}(\lambda)$ is the computable function on $\mathbb{N}^{4}$ given by the cited proposition. This is presented as a conjectural corollary for $\mathrm{F}_{4}$-type geometric representations; its resolution is not specified in the source.