Lukyanov’s conjecture (2000) states that in the Sinh-Gordon QFT in 1+1 dimension and finite volume, one can recover the matrix element of the exponential of the field operator via the study of a certain N-fold integral, Z_N. This question motivated the study of the so-called sinh-model, which is the probability whose partition is exactly Z_N. This ensemble is very similar to the famous β-ensemble in random matrix theory. Thanks to this ressemblance, the most important properties of the integral were studied in [Borot, Guionnet, Kozlowski 16’] in a slightly simpler case. In joint works, with Karol Kozlowski (ENS Lyon), we consider the situation corresponding to the study of Lukyanov’s conjecture and manage to give a strong check on its validity. Our set of techniques include Riemann-Hilbert problems and biorthogonal polynomial.