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Under normal conditions, I am not certain, as per Atomistry on $\ce{ZnCO3}$, the following quote:

The hemihydrate, $\ce{2ZnCO3.H2O}$, was obtained by precipitating the solution of a zinc salt with ammonium bicarbonate and digesting the gelatinous precipitate in excess of the precipitant till it became crystalline.

where I would have expected zinc bicarbonate, and further:

The basic zinc carbonate, $\ce{5ZnO.2CO2.4H2O}$, can be prepared by the direct hydrolysis of the normal carbonate, or by boiling a solution of zinc, zinc hydroxide, or zinc carbonate in an excess of aqueous carbonic acid.

And finally, a reference to a presumably formed zinc bicarbonate, and at times, citing the employment of pressure:

Basic zinc carbonates can be converted into the normal carbonate by contact with water charged with carbon dioxide under pressure. The normal carbonate is itself somewhat soluble in water containing carbon dioxide - presumably forming the bicarbonate.

The last quote is unclear, in my opinion, as to whether under normal experimental conditions, should one expect the creation of a stable $\ce{Zn(HCO3)2}$?

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We have to distinguish between the bicarbonate in solution and the bicarbonate as a solid. While many metal cations are compatible with bicarbonate anions in dilute aqueous solution, only in a few cases can a solid bicarbonate be expected as a precipitate. In most cases, such as with lithium or with hard-water forming metals such as calcium and magnesium, the carbonate is formed instead. The only solid metal bicarbonates I know of are those of the heavier alkali metals than lithium.

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