I haven't ever come across such a claim, if we require that it should refer to the classes that today are called "primitive recursive", respectively "total recursive". Indeed the class of functions we today call "primitive recursive" is usually cited as having been first defined in by Gödel in 1931 (Über formal unentscheidbare Sätze), which was after Ackermann had exhibited a function that is total and computable but not primitive recursive.
Of course it is conceivable, as Hilbert, Ackermann, et al. were struggling in the 1920s to find a suitable formalization of mechanical procedures, that somebody at some time considered whether something equivalent to primitive recursion would be enough to do the trick -- but whether this ever rose to the level of a "conjecture" might me more of a question of definitions than of history.
However, note that Gödel himself in 1931 when he defined the functions we know as primitive recursive, called them simply "recursive". It was only later that consensus decided, somehow, that "recursive" should mean what it does now. But that was merely a case of terminology changing, not the mathematical substance.
Later: See also here (from a book review written by Kleene in 1952):
Hilbert's lecture Über das Unendliche (published 1926) ... proposed to attack the continuum problem of set theory by showing that there is no inconsistency in supposing that the number-theoretic functions are all definable by use of forms of recursion associated with the transfinite ordinals of Cantor's second number class. ... For Hilbert's proposal it was necessary to show that higher forms of recursion do give new functions; and the first demonstration of the existence of a function definable by a double recursion but not by use only of simple or "primitive" recursion was given by Ackermann in 1928 ...
Ackermann's 1928 article works with a concept that looks to be fairly close to primitive recursion. Why Gödel is usually credited with inventing primitive recursion three years later, is then not completely clear. Perhaps just because Gödel made more heroic use of them, showing that they can do rather interesting things rather than just be the base layer in a hierarchy.
But this history seems to make clear that it probably wasn't seriously proposed that primitive recursion would be sufficient to express everything computable. It is unclear that computability was on anyone's mind in this context before 1931. In fact, the secondary sources I've been able to google up all claim that the idea that "computable function" is a robust concept at all did not congeal until the mid-1930s at the earliest, so in 1928 nobody would have even had the vocabulary to conjecture that the primitive recursive functions exhaust the computable ones.