Is Sigma 50 mm f/1.4 DG HSM Art compatibile with Nikon Z50?

TL;DR: Very little has been said about Sigma lenses used in conjunction with the FTZ adapter + Z50, but you'll probably be okay.

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Sigma has published lens compatibility statements for their recent (post- November 2013) lenses mounted on the Nikon FTZ (F-mount to Z-mount) adapter, for use on Nikon's Z-mount camera bodies. Note that these compatibility statements were released for the Z7, and later the Z6; Sigma has not issued a specific statement regarding the Z50.

However, the Z6 statement essentially said that their lenses behave identically between the Z6 and Z7, so the compatibility list is basically the same. Unfortunately, there seems be very few reviews of the Z50 used in conjunction with Sigma Art/Sport/Contemporary lenses. This review (Review: Nikon Z50) uses, amongst other Nikon F-mount lenses, the Sigma 135mm F1.8 Art and 14-24mm F2.8 Art lenses. The reviewer didn't note any problems.

Generally, I doubt there will be much issue using the Sigma lenses with the FTZ + Z50. Ostensibly, that's the entire point of Sigma's upgradeable firmware feature using their USB dock: issues can be patched with firmware updates.

Suggestion: if possible, try renting the equipment you're interested in before buying. Often, the rental houses (such as LensRentals.com and BorrowLenses in the US) have vast knowledge and expertise, and will be able to let you know if they expect any compatibility issues with the gear you're interested in.

Sigma's original compatibility statement had the following to say with regards to the 50mm F1.4 DG HSM Art when used with FTZ + Z7:

Phenomena particular to 4 lenses were confirmed after our announcement of September 28th: After careful investigation, we confirmed some phenomena particular to 4 products listed below. Regarding a firmware update for the SIGMA 24-35mm F2 DG HSM | Art, 50mm F1.4 DG HSM | Art and 85mm F1.4 DG HSM | Art, we are going to announce them at a later date.

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• SIGMA 50mm F1.4 DG HSM | Art
  [Phenomenon]
  When starting to shoot with the subject completely out of focus, the response to the AF operation is intermittent. It is necessary to release several times or to turn the focus ring once to release.
  [Future Course of Action]
  It is planned to be resolved by a firmware update.

The January 24, 2019 update addressed this:

[Applicable product]

• SIGMA 50mm F1.4 DG HSM | Art for NIKON F mount

[Benefit of the update]
When the lens is used in combination with the Nikon “Z 7” and their “Mount Adapter FTZ”, it has corrected the phenomenon whereby the response to the AF operation is intermittent when starting to shoot with the subject completely out of focus.

Sigma's statements:

• Operating conditions of the Nikon “Z7” and SIGMA’s interchangeable lenses for Nikon mount (October 26, 2018)
• Operating conditions of the Nikon “Z 6” and SIGMA’s interchangeable lenses for NIKON F mount (February 21, 2019)
• Firmware update for SIGMA’s interchangeable lenses for NIKON F mount (January 24, 2019)