The problem is more deeper. Rajab being one among the sacred months itself could well be a conspiracy.
The Quran states the following about the scared months.
Lo! the number of the months with Allah is twelve months by Allah's ordinance in the day that He created the heavens and the earth. Four of them are sacred: that is the right religion. So wrong not yourselves in them 9:36. The very next verse 9:37 also warns us about altering the sacred months and mentioned it as a sign of kufr. The Quran mentions in the very same verse the habit of Mushriks altering the months and makes strong objection to their action saying that ‘the evil of their doings is made fair-seeming to them’.
So there are 12 months and 4 of them are sacred. Muslims are supposed to meticulously respect this period, allow freedom of movement for everyone, and not supposed to wage war during scared months (unless they are attacked). Now let us focus on verse 9:1-3 for a closer understanding.
بَرَاءَةٌ مِنَ اللَّهِ وَرَسُولِهِ إِلَى الَّذِينَ عَاهَدْتُمْ مِنَ الْمُشْرِكِينَ فَسِيحُوا فِي الْأَرْضِ أَرْبَعَةَ أَشْهُرٍ
[This is a declaration of] disassociation, from Allah and His Messenger, to those with whom you had made a treaty among the polytheists. [Announce unto them:] "Go, then, freely about the earth for four months"
It is clear that it is the very same 4 sacred months which is mentioned in the above verse. The Muslims were directed to respect the free movement of their enemies. The next verse goes on to state another declaration which is made on the ‘yamul hajjul akbar’.
وَأَذَانٌ مِنَ اللَّهِ وَرَسُولِهِ إِلَى النَّاسِ يَوْمَ الْحَجِّ الْأَكْبَرِ
"And [it is] an announcement from Allah and His Messenger to the people on the day of the greater pilgrimage"
We can conclude that the above these;
1. Since Mushriqeen are free to roam freely on the earth for four months, naturally they are to be four consecutive months, considering also that, it states in verse 9:5; “Then when the Hurum Months have “insalakha” (passed in succession), you may kill the Mushrikeen wherever you find them. Seize them and besiege them and lie in wait for them at every point.” 9:5. The key word in the above verse and which is relevant to this matter is the word “insalakha”. The word literally means “skinned”. This is the sense of how we skin a snake for example. The movement of skinning a snake means we would remove the skin gradually from one end to the other, or in other words; a gradual and consecutive movement. Allah could have simply said: when the (hurum) months have passed. So why did Allah deliberately uses this word? It is because the word “insalakha” indicates a consecutive meaning, or in other words it tells us that the 4 hurum months are consecutive months.
2. Mushriks are prohibited to approach the Sacred Mosque after this year of theirs…” [9:28] and therefore, the permission granted for free movement does not include Muharram, Safer and Rahihul Awwal and Akhir.
Based on the above, Yamul hajjul akbar is the annoucement made on the first day of Ramadan. It is pertinent to note that Quran has mentioned al hajj shahrun (three or more months) malomat (2:197) The Hajj shall be observed during the well-known months. It may be that Hajj involves great effort during those times, those who undertakes the Hajj during that time may try to reach before the advent of Ramadan.
While we under Yamul hajjul akbar is the first day of Ramadan the 4 consecutive months ends after Dul Hajj, which is the end of the year (Ramadan, Shawwal, Dul Qadah and Dul Hajj). There is no logical explanation why the month of Ramadan is not a scared month.
May Allah show us the right path. Alhamduillah always