Mechanism for Diels–Alder of alkene with substituted thiophene sulfone with loss of sulfur dioxide
A few references into the rabbit hole and we have an answer. The major reference was given in the link mentioned by OP which was Chem. Ber. 1994, 127 (2), 409–425[1], in which the reaction mechanism of the synthesis is given as follows:


[...] Reaction of 33 with tetrathiophene-S,S-dioxide is performed by heating the reactants in highly concentrated toluene solution to $\pu{120 °C}$ for a total of
24 h. The cycloadduct 34 is obtained in high yield.

Now the question that has arisen in the step from 33 to 34 is the following, under normal conditions of Diels-Alder, there would only be one double bond in the final product. Here there are two. This can be answered going one down into the rabbit hole and using J. Org. Chem. 1980, 45 (5), 856–867.[2]
This paper deals with the specific reaction that causes confusion - the annulation using tetrachlorothiophene 1,1-dioxide. According to this paper:

The compound has a combination of stability and reactivity that makes it a useful synthetic reagent. It undergoes Diels-Alder addition with double bonds to form an adduct which spontaneously loses sulfur dioxide to leave the double bond annelated with a tetrachlorobutadienediyl group.


Furthermore, it talks about the reactivity to olefinic bonds and states the following:


Example 18, from l-hexen-5-yne, reveals the greater reactivity of alkene vs. alkyne. Reaction is exclusively with the alkene bond as far as NMR reveals. Thus, the kinetically favored product is formed rather than the thermodynamically favored one.

There is a minor error in the reaction mechanism as given in the link of OP as the first paper shows that the tetra-chloro derivative needs to be oxidised in order to be converted into cyclohexaindane
References:

Kuck, D. Benzoanellated Centropolyquinanes, 15. Benzoanellated Fenestranes with [5.5.5], [5.5.5.6], and [5.5.5.5] Frameworks: The Route from 1,3-Indandione to Fenestrindan. Chem. Ber. 1994, 127 (2), 409–425. DOI: 10.1002/cber.19941270218.
Raasch, M. S. Annulations with tetrachlorothiophene 1,1-dioxide. J. Org. Chem. 1980, 45 (5), 856–867. DOI: 10.1021/jo01293a019.