On the standard conjecture for a $3$-dimensional variety fibred by curves with a non-injective Kodaira–Spencer map

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a complex projective 3-dimensional variety fibred by curves (possibly with degeneracies) over a smooth projective surface provided that the endomorphism ring of the Jacobian variety of some smooth fibre coincides with the ring of integers and the corresponding Kodaira–Spencer map has rank $1$ on some non-empty open subset of the surface. When the generic fibre of the structure morphism is of genus $2$, the condition on the endomorphisms of the Jacobian may be omitted.