Is every semigroup with (possibly non-unique) division a group?

Let's say that a semigroup $(S,\cdot)$ has weak division if for all $a,b\in S$ there exist $c,d\in S$ such that $ac = b$ and $da = b$. Note that we don't require $c$ and $d$ to be unique. This property may already have a name in the semigroup literature, but some google and wikipedia browsing didn't turn up anything.

In this question, it was asked whether every commutative semigroup $(S,+)$ with weak division is a group. In my answer, I pointed out that it's true if we assume uniqueness, since then we have an associative quasigroup, and further that existence implies uniqueness for finite semigroups, so any counterexample must be infinite.

My answer to that question has been accepted, and my hunch is that commutativity shouldn't be very important here, so I thought I'd ask the more general question.

Is every semigroup with weak division a group? If not, is it true for commutative semigroups?

Edit: I've just become aware of this question, which is this same, but only for finite semigroups. I don't think it's a duplicate, since the answer given there uses finiteness to get an idempotent element and then applies Green's theorem. So it seems that any counterexample must contain no idempotent.

The answer is yes (assuming that $S$ is nonempty). Let $s \in S$. Then there exists $e$ such that $es = s$. Then $e$ is idempotent. Indeed, there exists $d$ such that $e = sd$. Therefore, $ee = esd = sd = e$.
Now, we are back to this answer: $S$ is equal to the $\mathcal{H}$-class of $e$, and an $\mathcal{H}$-class containing an idempotent is a group.