Grading: Its chain is "da'if" because Sulayman and his father are unknown (as al-Dhahabi said in al-Talkhis).
Technical note / Defect ('illah): Ibn Qudamah (al-Muntakhab min 'Ilal al-Khallal: 137) reported from Muhanna al-Shami that he asked Yahya bin Ma'in about Sulayman bin Abi Sulayman and mentioned this hadith, and Yahya said: "We do not know him" (i.e. Sulayman). Muhanna said: And Ahmad bin Hanbal said to me: "The well-known students of Abu Hurayrah do not have this hadith." Ibn al-Jawzi (al-'Ilal al-Mutanahiyah: 1277) mentioned it and said: This is not sound. And Ibn Kathir (al-Bidayah wa al-Nihayah: 11/ 114) also mentioned it and said: It is "very gharib".

Reference / Source: Al-Bayhaqi (Dala'il al-Nubuwwah 6/ 447), Ibn 'Abd al-Barr (Jami' Bayan al-'Ilm 2324), al-Khatib (al-Muttafiq wa al-Muftariq 626), al-Daylami (Musnad al-Firdaws, as cited in al-Ghara'ib al-Multaqatah of Ibn Hajar 1578) and Ibn 'Asakir (Tarikh Dimashq 1/ 183-184) recorded it via Hushaym bin Bashir with this same chain. And al-Bukhari mentioned it (al-Tarikh al-Kabir 4/ 16) via 'Amr bin 'Awn > Hushaym. And Ibn 'Asakir (1/ 183) also recorded it via Muhammad bin al-Hasan bin 'Imran al-Wasiti > al-'Awwam bin

Note / Clarification: (continued) Hawshab, with this same chain.

Reference / Source: Nu'aym bin Hammad recorded it in "al-Fitan" (248) via Muhammad bin Yazid al-Wasiti > al-'Awwam bin Hawshab > a man (unnamed) > Abu Hurayrah as "mawquf" (as Abu Hurayrah's own words).
Important point: And this is closer to what is correct. And Allah the Exalted knows best.

Corroborating chains and supporting reports: It has a supporting report in the hadith of 'Abd al-Malik bin 'Umayr (from someone who narrated it to him from the Messenger of Allah (peace and blessings be upon him)) in Ibn 'Asakir (1/ 185), with the wording: "My caliphate will be in Madinah and my kingdom in al-Sham." This is via Hisham bin 'Ammar > Shihab bin Khirash > 'Abd al-Malik bin 'Umayr.
Technical note / Defect ('illah): Hisham is saduq (truthful), but in old age he began to accept prompting (talqin). A group declared Shihab bin Khirash strong, but Ibn 'Adi said some of his narrations are munkar, and Ibn Hibban said he made many mistakes. 'Abd al-Malik bin 'Umayr's memory deteriorated and he sometimes committed tadlis. Therefore this chain is "da'if" because of Hisham bin 'Ammar being alone in it and because 'Abd al-Malik's source is unknown.