As it was pointed out in one answer of this question, the BGV paper has a footnote that says
One can obtain a "pure" FHE scheme (with a constant-size public key) from these leveled FHE schemes by assuming "circular security", namely that it is "safe" to encrypt the leveled FHE secret key under its own public key.
However, I do not understand how the circular security assumption is related to this transformation from a leveled FHE to a pure FHE.
In my understanding, any leveled FHE can be turned in a pure FHE by setting the acceptable multiplicative depth, that is, the parameter $L$, to a value large enough to evaluate homomorphically the decryption circuit plus one operation and then apply the Gentry's blueprint, namely, refresh the ciphertexts after any operation by running the decryption function homomorphically.
So, can anyone explain what authors of BGV mean with that footnote or show what is wrong with my understanding ?