  Given proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$,
respectively, an $m \times n$ matrix $A$ with real entries is said to be
semipositive if there exists a $x \in K_1^{\circ}$ such that $Ax \in
K_2^{\circ}$, where $K^{\circ}$ denotes the interior of a proper cone $K$. This
set is denoted by $S(K_1,K_2)$. We resolve a recent conjecture on the structure
of into linear preservers of $S(\mathbb{R}^n_+,\mathbb{R}^m_+)$. We also
determine linear preservers of the set $S(K_1,K_2)$ for arbitrary proper cones
$K_1$ and $K_2$. Preservers of the subclass of those elements of $S(K_1,K_2)$
with a $(K_2,K_1)$-nonnegative left inverse as well as connections between
strong linear preservers of $S(K_1,K_2)$ with other linear preserver problems
are considered.
