with all items geared up ,the trinket is not working properly.
as it reported before ,
the first 2 tick add 0 spell damage ,the 3rd adds 40 ,4th adds 80 ,5th adds 120 ,6th adds 0 ,
while it should work as 1st add 40 ,2se add 80 ,3rd add 120 ,4th add 160 ,5th add 200 ,6th add 200.

I use a command that removes the damage spread and it is always static depending on spell damage. For example Smite 9 rank 384 to 429 but I always deal 383 damage.
So you may not notice the damage increase on 1 and 2 ticks.
in the screenshot we can see how the last 5 spells get a damage boost. And on the 6th spell, it resets according to the description.

but yes, we have a 1 step discrepancy according to this comment

real issue found :
this trinket has conflict with priest's spell "inner fire"(talent "improved inner fire" is on )

look without inner fire buff ,it works fine .

with inner fire on ,mess up the effect of the trinket

the conflict between inner fire and this trinket is
the first 2 tick become no effect ,and the 3rd ,4th ,5th act like there is no innner fire buff.

first 2 tick add 40 and 80 ,< 94 improved inner fire , so the system choose 94
3rd 4th 5th add 120 160 200 , >94 ,so the system choose the trinket .

but they should be stack.

well yea, seems aura 14770 [Improved Inner Fire] (spell 14747,14770 [Improved Inner Fire, rank 1 enUS]) doesn't stuck with aura 28204 [Ascendance]
item 22678
statinfo shows:
use inner fire - 58SP
use trinket and 1 cast - 58 SP
second cast - 80 SP

About the 6th tick ,maybe blizzard found it too op then just code it in the wrong way, I mean whats different of following:

Expires after 6 damage or healing spells or 20 seconds, whichever occurs first.
Expires when 6 damage or healing spells or 20 seconds, whichever occurs first.
Expires after 5 damage or healing spells or 20 seconds, whichever occurs first.

"Your next 5 damage or healing spells cast within 20 seconds "
I dont cast anything , use the trinket ,then cast a spell ,is this spell count as the first of "Your next 5"?

So , i dont think :
Here's how the trinket works:

-Activate trinket
-1st spell: 0 bonus spell damage
-2nd spell: 40 bonus spell damage
-3rd spell: 80 bonus spell damage
-4th spell: 120 bonus spell damage
-5th spell: 160 bonus spell damage
-6th spell: 200 bonus spell damage
-Buff fades now or 20 seconds after activation, whichever is first.

match the words of its description.

But continuing 2 200 spell damage bonus is kind of op ,adjust is reasonable.

without the talent "improved inner fire" ,the result remain the same ,
the aura "inner fire" and the trinket just conflict each other:
which the +spell damage larger the system choose that one aura.
But they should be stack.

first 1 tick add 40 ,< 73 inner fire , so the system choose 73
2sd 3rd 4th 5th add 80 120 160 200 , >73 ,so the system choose the trinket .
you can see the 2sd 80 - 73 =7 ,so the shadow word pain only increased 1 per tick.
clearly shows system only choose bigger one aura from inner fire and trinket.