Generic lines and one fat linear space have good postulation

Let $n,d,r\in\mathbb{N}$ with $n\geq r+2\geq 3$. Let $\Pi\cong\mathbb{P}^r\subset\mathbb{P}^n$ and let $X\subset\mathbb{P}^n$ consist of $s\geq 1$ generic lines and one $m$-multiple linear space $m\Pi$, where $m\geq 2$. A scheme has good postulation in degree $d$ when its hypersurfaces of degree $d$ impose the expected number of independent conditions. Fat-linear-space postulation conjecture. The scheme $X$ always has good postulation, except when $n=r+3$, $m=d$, and $2\leq s\leq d$. This stronger conjecture generalizes the multiple-line formulation and is motivated by the absence of known defective cases outside the stated numerical configuration; the paper proves only a special case.