Select different numbers from indipendent lists efficiently

Hello everybody. I'm looking for a volatile UDF that selects a random entry from every column in a given range so that every selected value is unique. The same numbers can appear in multiple columns, but not more than once in a column. As i stated, the UDF needs to update whenever i refresh the worksheet, wheter or not the input numbers change. The input parameter would be the data range (e.g. If the starting data was A:[1,2,5], B:[6,3,2], C:[4,5,6,7], D:[6], E:[3,7,8] the UDF's input would be A1:E4, so note that the input contains empty cells too) and the output would be an array containing a random solution (e.g. [1,3,7,6,8] or [3,2,4,6,7], with the previous data). The UDF needs to take an optimised path, otherwise it's going to be very slow with large datasets. This is probably the hardest part, which is why i asked for help in this forum.
Here is a sample to better understand the problem: samples.xlsm. My datased is substantially larger than this, which is why i need an efficient method.
--to the moderators, I'm aware that i've already asked this on another thread and other forums but the answer i got is giving issues so I posted again to state the problem more clearly and hopefull receive new answers. Please don't take this down, thank you--

see sheet "blad1", 1.000 rows and 10 columns filled with random values 1-10.
Output in columns M:V for those unique values, in column X the executiontime in secondes for 2 methods. (this time is sometimes too short)

how do i get rid of this part of the array Attachment 827607, keeping only the solution's numbers?. Other than that, the function works perfectly fine, this was incredibly helpful

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invalid attachment in #3.
There were 3 lines too much ...

Thank you. I have one last question. Would it be possible to modify the code for the UDF to still work with a formula that creates a 2D array (like the one that generates variable lists in the sample i gave) as the input?

like this ?

I think it's even faster.
My example with 1000 rows and 10 columns wasn't that good, because you said that every entry in a column is unique, but the columns have common numbers and the result has to have again unique numbers. If you have only 1 column with 1.000 records and the others for example only approx. 10, then i can better add some lines to speed that up.

you can use the last one (#10) in all situations. Good luck.