Lower bound for abelian squares from effective partitions

Let $w$ be a non-boundary binary word. For a positive integer, write $e(r)$ for the effective partition notation used in the source, and suppose

$$e(|w|_a)=[i,k],\qquad e(|w|_b)=[j,k].$$

An abelian square is a factor consisting of two consecutive factors with the same Parikh vector.

Effective-partition conjecture. The word $w$ contains at least

$$\left\lfloor \frac{i}{2}\right\rfloor+\left\lfloor \frac{j}{2}\right\rfloor$$

abelian squares.

The supplied context does not define the notation $e$ or explain whether this assertion is intended as a conjecture beyond the displayed candidate; the status is therefore left open.