 Solution

True. Coordinate projections define ${\pi }_1(X\times Y,(x,y))\to {\pi }_1(X,x)\times {\pi }_1(Y,y)$. A pair of based loops gives the inverse by $t\mapsto ({\gamma }_X(t),{\gamma }_Y(t))$. Homotopies combine and project coordinatewise, and loop concatenation does too. These maps are well-defined inverse group homomorphisms, requiring no local connectedness assumptions.