Lower-bound conjecture for convex-polytope f-vectors

Let $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ is the number of $k$-dimensional faces. The face-number lower-bound conjecture. For every $k$ with $0\le k\le d-1$,

$$f_k\ge \min\{f_0,f_{d-1}\}.$$

The source calls this a suspiciously innocuous conjecture and says that apparently no one had an idea for a proof at the time. It is presented as an open problem about the shape of convex-polytope $f$-vectors.