What is the relationship between Benjamin and Leibowitz?

In Fiat Homo, one of the monks of the Albertian Order of Leibowitz (Brother Fingo) finishes a hand-carved woodcarving of Leibowitz (who had not yet been deemed a saint).

The woodcarving was passed down from generation to generation, and winds up in the possession of Dom Zerchi, the abbot of the monastary over 1000 years later in the Fiat Voluntas Tua section.

There were repeated references in particular to the smile on the statue's face, and the implication that the smile was that of Benjamin, the seemingly immortal Jewish hermit.

Brother Francis, who is the first in the story to encounter Benjamin, remarks upon the similarity when he sees the smile start to emerge during the carving process:

But as the work progressed, Francis could not escape the feeling that the face of the carving was smiling a vaguely familiar smile. He sketched it thus, and the feeling of familiarity increased. Still, he could not place the face, or recall who had smiled so wryly.

And then again upon completion:

"I know him!" Francis gasped, staring at the merry-but-sad wrinkled eyes, the hint of a wry smile at the corners of the mouth - somehow almost too familiar.

"You do? Who is it then?" wondered Fingo.

"It's - well, I'm not sure. I think I know him. But-"

Fingo laughed. "You're just recognizing your own sketches," he offered in explanation.

Francis was not so certain. Still, he could not quite place the face.

Hmm-hnnn! the wry smile seemed to say.

Later on, Zerchi also reflects upon the resemblence:

The image was old, very old. Some earlier ruler of the abbey had sent it down to a basement storeroom to stand in dust and gloom while a dry-rot etched the wood, eating away the spring grain and leaving the summer grain so that the face seemed deeply lined. The saint wore a slightly satiric smile. Zerchi had rescued it from oblivion because of the smile.

"Did you see that old beggar in the refectory last night?" he asked irrelevantly, still peering curiously at the statue's smile.

"I didn't notice, Domne. Why?"

"Never mind, I guess I'm just imagining it."

The implication is that the image Fingo carved is that of Benjamin, rather than Leibowitz, or possibly that Benjamin is Leibowitz (which was also somewhat implied during Francis' encounter with Benjamin, and speculated upon widely within the monastery afterwards).

Benjamin himself stated in the Fiat Lux section that Leibowitz was a "distant relative":

"now, now, paulo. one of them once mistook me for a distant relative of mine - name of Leibowitz. He thought I had been sent to deliver him a message - or some of your other scalawags thought so. I don't want it to happen again, so I throw pebbles at them sometimes. Hah! I'll not be mistaken for that kinsman again, for he stopped being any kin of mine."

The priest looked puzzled. "Mistook you for whom? Saint Leibowitz? Now, Benjamin! You're going too far."

Benjamin repeated it in a mocking singsong: "Mistook me for a distant relative of mine - name of Leibowitz, so I throw pebbles at them."

Dom Paulo looked thoroughly perplexed. "Saint Leibowitz has been dead a dozen centuries. How could-" He broke off and peered warily at the old hermit. "Now, Benjamin, let's don't start that tale wagging again. You haven't lived twelve cent-"

So what is the relationship between Leibowitz and Benjamin? Is the woodcarving supposed to be of Leibowitz, or of Benjamin? Are they supposed to be one and the same, despite Benjamin's claims?