A suitable alternative identification strategy would be to implement a regression discontinuity design. Which raises the question, why do the authors define their treatment group as units built between $1900$ and $1980$?
$$ \underbrace{\lim {t \to 1980-}\mathbb{E}[Y{i} \vert T_i=t]}{\text{Treated}} - \underbrace{\lim {t \to 1980-}\mathbb{E}[Y{i} \vert T_i=t]}{\text{Control}} $$
