Weak coupling limit of KPZ with rougher than white noise

We consider the KPZ equation in $1$ spatial dimension with noise that is rougher than white by an exponent $\gamma >1\mathrm{/}4$. Under a weak coupling limit, formally removing the nonlinearity from the equation, we show using regularity structures that the renormalised solutions converge to a Gaussian limit that is different from the solution of the linear part of the equation. The regime of this effect has a nontrivial overlap with the subcritical regime $\gamma <1\mathrm{/}2$.