I will begin with a disclaimer that while I understand the general underlying principles behind PLS, my linear algebra background is rather limited. I have trouble with the details of constraints on the various elements of the method, which is the reason I'm asking this. I've taken a look at a number of papers reviewing the uses of PLS and answers on this site explaining PLS theory (such as this one), none addressing my issue
Say we are in the scenario of multi-response PLS (or PLS2). We have $Y$ the dependent values matrix $n$ observations by $p$ variables, $X$ the independent values matrix $n$ by $q$.
I'm aware that PLS deals rather well in instances with $n << p$, provided that $n$ itself isn't too small. Also, that if $q >> p$, we are likely to overfit.
My question however, lies more in whether there are issues if $q << p$.
As far as I'm concerned, PLS regression (PLSR) is very powerful in a multi-response scenario where there are highly correlated groups of variables within $Y$ (such as gene expression data for instance).