The spanning conjecture for f-vectors of ordinary polytopes

Let $d$ be an odd integer with $d\ge 5$. An ordinary $d$-polytope is a polytope whose face structure is governed by the ordinary-polytope construction; write $f_i$ for the number of its $i$-dimensional faces. The Euler hyperplane is the affine hyperplane

$$\sum_{i=0}^{d-1}f_i=2.$$

Spanning conjecture. The set of $f$-vectors of all ordinary $d$-polytopes spans the Euler hyperplane. A spanning set consists of the ordinary polytopes

$$P^{d,d+\lfloor i/2\rfloor,d+i},\qquad 1\le i\le d.$$

This predicts that these explicitly specified ordinary polytopes provide enough $f$-vectors to span all admissible directions in the Euler hyperplane. The supplied text gives no resolution, so the conjecture is left open.