Gee--Herzig--Savitt's tame weight conjecture

Let $F$, $\chi$, and $\overline r$ be as in the paper, with $\overline r$ automorphic of some weight and level. Let $W(\overline r)$ be its set of modular Serre weights, and let $W^?(\overline r_p|_{I_{\mathbf Q_p}})$ be the predicted Serre-weight set. Gee--Herzig--Savitt's conjecture. If $\overline r|_{G_{F_v}}$ is tame and sufficiently generic at every $v\mid p$, then

$$W(\overline r)=W^?(\overline r_p|_{I_{\mathbf Q_p}}).$$

The source attributes this conjecture to Gee--Herzig--Savitt and applies its $\mathrm{GSp}_4$ analogue under suitable genericity and Taylor--Wiles hypotheses.