Maximum degree of noninvertibility for promotion on posets

Let $P$ be an $n$-element poset, and let $\partial$ denote promotion on labelings of $P$. Write $\deg(\partial)$ for its degree of noninvertibility.

Maximum noninvertibility conjecture. For any poset $P$,

$$\deg(\partial)\leq\frac{(n+2)(n+1)}{6}.$$

In other words, the degree of noninvertibility of promotion should be largest when $P$ is a chain. The stated value is attained for an $n$-element chain, while the general inequality remains open.