Conjecture on the trace description of the unit for closed symmetric semigroup categories

Let $\mathcal{S}$ be a closed symmetric semigroup category. Consider the category of functors

$$[\mathcal{S}^{\operatorname{op}},\mathbf{Vec}_{\Bbbk}].$$

Trace-unit conjecture. The category $[\mathcal{S}^{\operatorname{op}},\mathbf{Vec}_{\Bbbk}]$ is symmetric monoidal, and its unit is equivalent to the trace of the underlying semigroup category:

\mathbb{1}_{[\mathcal{S}^{\operatorname{op}},\mathbf{Vec}_{\Bbbk}]} \simeq \operatorname{Tr}_{\mathcal{S}}(\left\llbracket \mathcal{S}\right\rrbracket).

The conjecture extends the preceding theorem from rigid symmetric semigroup categories to all closed symmetric semigroup categories, asserting that the trace still describes the monoidal unit when rigidity is not assumed.