Let $O$ be an operad in $\mathtt{SETS}$. Assume that $O(0)$ is empty and $O(1)$ only consists of the identity. Assume for simplicity that $O$ is monochromatic, i.e. we have no labels on the in/outputs. Assume also for simplicity that the operad is plain, i.e. neither symmetric nor braided. So the operads in question consist of a set of $n$-ary operations $O(n)$ for each $n\in\mathbb{N}$ together with an associative composition and there is a unit element in $O(1)$ (but no more elements, as required above).

Now $O$ freely generates a monoidal category $S(O)$: The objects are natural numbers and an arrow from $m$ to $n$ consists of a sequence of operations in $O$ with a total of $m$ inputs and a total of $n$ outputs. For example if $a\in O(3)$ and $b\in O(5)$, then $(a,b)$ is an arrow from $3+5=8$ to $2$. Composition is given by composition in the operad.

I know that $S(O)$ is aspherical when $O$ is free and also in some other special cases. Here I consider categories as spaces via the usual geometric realization, i.e. the geometric realization of the nerve of the category.

Question: Is the category $S(O)$ always aspherical?