Non-closure conjecture for invertible elements in Bol magmas

Let $L$ be a Bol magma with a neutral element, and let $\mathcal{J}(L)$ denote the set of elements of $L$ possessing a two-sided inverse. Non-closure conjecture. There exists a Bol magma $L$ with a neutral element such that $\mathcal{J}(L)$ is not closed under multiplication. This would show that the preceding loop-closure results for special classes of Bol magmas do not hold for arbitrary Bol magmas; the source provides no resolution.