  We provide a description of the quantum integrable structure behind the
Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and
Shatashvili (NS) for $\mathcal{N}=2$ 4d Super Yang-Mills (SYM) theories. In
this regime of the background, -- we shall show --, the instanton partition
function is characterised by the solution of a TQ-equation. Exploiting a
symmetry of the contour integrals expressing the partition function, we derive
a 'dual' TQ-equation, sharing the same T-polynomial with the former. This fact
allows us to evaluate to $1$ the quantum Wronskian of two dual solutions (for
$Q$) and, then, to reproduce the NS TBA-like equation. The latter acquires
interestingly the deep meaning of a known object in integrability theory, as
its two second determinations give the usual non-linear integral equations
(nlies) derived from the 'dual' Bethe Ansatz equations.
