Failure of the weak Lefschetz property for almost complete intersections of squares

Let $A_{r,2}$ denote the quotient algebra formed from $r$ generic linear forms in the construction used in the paper, with the generators squared, and let WLP denote the weak Lefschetz property. Almost-complete-intersection squares conjecture. The algebra $A_{r,2}$ fails WLP for

$$r=6\quad\text{and all}\quad r\ge 8.$$

The claim is motivated by dimension computations and numerical experiments for the maps in consecutive middle degrees. The paper gives examples and partial criteria but does not establish the asserted range in full.