Heterochromatic number conjecture for circuits in projective geometries

Let $PG(r-1,q)$ be the projective geometry of rank $r$ over the finite field with $q$ elements, and let $C_p(PG(r-1,q))$ be the hypergraph whose hyperedges are the $p$-circuits of this matroid. The heterochromatic number $hc(H)$ of a non-empty hypergraph $H$ is the smallest integer $k$ such that every colouring of its vertices with exactly $k$ colours contains a totally multicoloured hyperedge. Heterochromatic projective-geometry conjecture. Given integers $3\leq p\leq r+1$ and a prime power $q$, the heterochromatic number is

$$hc\bigl(C_p(PG(r-1,q))\bigr)=\frac{q^{p-2}-1}{q-1}+(r-p+1)+1.$$

The statement arises from a construction using a maximal flat without a $p$-circuit and a colouring of the corresponding contraction. The supplied text does not establish whether this proposed formula is proved or remains open.