Invariance of density cardinals under changes outside the open unit interval

Let ${\mathsf{all}}$ be the ambient parameter set, and let $X,X'\subseteq{\mathsf{all}}$ satisfy

$$X\setminus(0,1)=X'\setminus(0,1).$$

Let $Y$ be arbitrary, and write ${\mathfrak{dd}}_{X,Y}$ for the associated density cardinal. Density-cardinal invariance conjecture. If $X$ and $X'$ agree outside $(0,1)$, then

$${\mathfrak{dd}}_{X,Y}={\mathfrak{dd}}_{X',Y}.$$

This is presented as a more sweeping conjecture following the singleton-target conjecture, and the paper does not provide a proof or resolution.