Exactness and full faithfulness of the convolution Hecke functor

Let $\Catgeomdk$ be the geometric category defined in the surrounding text, let $\Pervsmkb$ be the stated target category, and let $\ConvHecke$ be the functor constructed there. Convolution Hecke-functor conjecture. The functor

$$\ConvHecke:\Catgeomdk\to \Pervsmkb$$

is exact and fully faithful. This would identify the geometric category with its image in the target and strengthen the preceding right-exactness result; the supplied text gives no resolution or supporting evidence.