Specht property for the $2$-graded identities of $B_m$

Let $K$ be a field of characteristic zero and $V$ a vector space of dimension $m>1$ with a nondegenerate symmetric bilinear form $f:V\times V\to K$. The Jordan algebra $B_m=K\oplus V$ of the form $f$ is a superalgebra with this decomposition. We prove that the ideal of all the $2$-graded identities of $B_m$ satisfies the Specht property and we compute the $2$-graded cocharacter sequence of $B_m$.