  In this paper we construct an infinite family of paramodular forms of weight
$2$ which are simultaneously Borcherds products and additive Jacobi lifts. This
proves an important part of the theta-block conjecture of Gritsenko--Poor--Yuen
(2013) related to the only known infinite series of theta-blocks of weight $2$
and $q$-order $1$. We also consider some applications of this result.
