Claims

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1965_01_01_moser: Moser's 1965 proceedings paper proves that every additive complement of the squares has liminf greater than 1.06, the first yes to the liminf question; the proceedings volume is not shown to be refereed, so the claim is pending.

1993_07_01_cilleruelo: Cilleruelo's 1993 theorem bounds a set completing the squares up to N below by (4/pi + o(1)) times the square root of N, so every additive complement of the squares has liminf at least 4/pi; it answers the liminf question yes.

1995_03_01_habsieger: Habsieger's 1995 theorem bounds an additive complement of a polynomial's values up to N below by an explicit constant, 4/pi for the squares, so every additive complement of the squares has liminf at least 4/pi.

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