An Extension of the Collatz Conjecture modulo $2^p+2^q$

In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new conjecture modulo 10 which may be stated as follows. Starting from any positive integer, if it is a multiple of 10 then divide it by 10, otherwise, multiply it by 12, add 8 times its last digit and divide the result by 10. Repeat the process infinitely. Regardless the starting number, the process eventually reaches 4 after a finite number of iterations. The genaral conjecture studied here will encompasse the classical Collatz conjecture togher with our proposed one modulo 10.