  In this article, we propose the explicit studies of $\hbar$-deformed
meromorphic connections in $\mathfrak{gl}_3(\mathbb{C})$ with an unramified
irregular pole at infinity of order $r_\infty=3$ and its spectral dual
corresponding to the $\mathfrak{gl}_2(\mathbb{C})$ Painlev\'{e} $4$ Lax pair
with one vanishing monodromy at the finite pole. Using the apparent
singularities and their dual partners on the spectral curves as Darboux
coordinates, we obtain the Hamiltonian evolutions, their reductions to only one
non-trivial direction, the Jimbo-Miwa-Ueno tau-functions, the fundamental
symplectic two-forms and some associated Hermitian matrix models on both sides.
We then prove that the spectral duality connecting both sides extends to all
these aspects as an explicit illustration of generalized Harnad's duality. We
finally propose a conjecture relating the Jimbo-Miwa-Ueno differential as the
$\hbar=0$ evaluation of the Hamiltonian differential in these Darboux
coordinates that could provide insights on the geometric interpretation of the
$\hbar$ formal parameter.
