Kelly's conjecture on reconstructibility from vertex-deleted decks

Let $l\in\mathbb{N}$. An $l$-card of a graph is an induced subgraph obtained by deleting $l$ vertices, and the corresponding deck is the multiset of all such cards. A graph is $l$-reconstructible if it is determined by this deck.

Kelly's conjecture. For $l\in\mathbb{N}$, there is an integer $M_l$ such that any graph with at least $M_l$ vertices is reconstructible from its deck of cards obtained by deleting $l$ vertices.

This is a more detailed version of the Reconstruction Conjecture. The original conjecture is the special case $M_1=3$; the asserted existence of a threshold for every $l$ remains open.