Hofer-small exact isotopy preserves unobstructedness

Let $L$ be an unobstructed Lagrangian with a bounding cochain of valuation $c$, and let $L'$ be exactly homotopic to $L$ through an exact homotopy whose Hofer norm is less than $c$. Hofer-small unobstructedness conjecture. Then $L'$ is unobstructed. The conjecture would ensure that the exact-homotopy decompositions used for Lagrangian cobordisms preserve unobstructedness; the source gives no resolution.