It is very tough. Read this:
Article 2 The Fund shall report to the Council of Economic and Development Affairs. It shall have a public legal personality as well as financial and administrative independence. The Fund's headquarters shall be in the city of Riyadh; it may, as needed, establish branches within the Kingdom or abroad
So it is independent but yet reports to the Council of Economic and Development Affairs (CEDA). The Crown Prince Mohammad bin Salman Al Saud presides over CEDA and its 22 members include heads of major ministries and some of the members of the Council of Ministers
Then this:
The Fund shall have a board of directors to be chaired by the President of the Council of Economic and Development Affairs . Its membership shall include the Governor, as well as a minimum of four experts and specialists, provided they include representatives from relevant agencies; they shall be appointed pursuant to a royal order for a renewable term of five years.
Funnily enough the experts are largely also ministers:
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9 board members - 7 of which are ministers.