The f-vector conjecture for tropical linear spaces

A tropical $d$-plane in $n$-space is a tropical linear space of dimension $d$ in $n$-space. Let $i$ be a face dimension, and map the space to the quotient by the diagonal line:

$$\mathbb{R}^n/(1,\ldots,1).$$

A face is called bounded when its image under this map is bounded. The f-vector conjecture. The number of $i$-dimensional faces that become bounded is at most

$$\binom{n-2i}{d-i}\binom{n-i-1}{i-1}.$$

There is theoretical and experimental evidence for this bound. The paper proves that every constructible tropical linear space achieves the bound, while the assertion for all tropical linear spaces remains open.