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.