Multicritical points of unitary matrix model with logarithmic potential identified with Argyres-Douglas points

In [arXiv:1805.05057 [hep-th]],[arXiv:1812.00811 [hep-th]], the partition function of the Gross-Witten-Wadia unitary matrix model with the logarithmic term has been identified with the $\tau$ function of a certain Painlev'{e} system, and the double scaling limit of the associated discrete Painlev'{e} equation to the critical point provides us with the Painlev'{e} II equation. This limit captures the critical behavior of the $su(2)$, $N_f=2$ $\mathcal{N}=2$ supersymmetric gauge theory around its Argyres-Douglas $4D$ superconformal point. Here, we consider further extension of the model that contains the $k$-th multicritical point and that is to be identified with ${\widehat{A}}_{2k,2k}$ theory. In the $k=2$ case, we derive a system of two ODEs for the scaling functions to the free energy, the time variable being the scaled total mass and make a consistency check on the spectral curve on this matrix model.