Asymmetry of $\mathbb{P}$-Functors

Recently, a new definition of $\mathbb{P}$-functors was proposed by Anno and Logvinenko. In their article, the authors wonder whether this notion is symmetric in the sense that the adjoints of $\mathbb{P}$-functors are again $\mathbb{P}$-functors, the analogue being true for spherical functors. We give geometric examples involving the Hilbert scheme of points on a surface that yield a negative answer.